Sophie Diagrams

The diagram attached consists of a rectangle ABCD and a triangle DXY so that X and Y are points on the line segments AB and BC respectively and angle DXY = 90 degrees. If all the line segments in the diagram have integer lengths, then we call it a Sophie Diagram.

We now want to find AB. However, there is only one pythagorean triple with hypotenuse 15, (9, 12, 15). We therefore need to find out if BY is 9 or 12. This is equivalent to showing whether AB=CD=12+9=21 or 12+12=24

To do this, note that DY = 25 by Pythagoras (the triple (15, 20, 25) ) and note that there are only two triples with hypotenuse 25,

(7, 24, 25) and (15, 20, 25).

Clearly, as 21 is not in either of these triples AB=CD=24, and we are done.